Embedded newform 322.2.m.a.179.8

• Label: 322.2.m.a.179.8
• Level: $322$
• Weight: $2$
• Character: 322.179
• Analytic conductor: $2.571$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	322.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.57118294509\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(8\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			179.8
Character	\(\chi\)	\(=\)	322.179
Dual form			322.2.m.a.9.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.235759 - 0.971812i) q^{2} +(0.902469 + 2.60751i) q^{3} +(-0.888835 + 0.458227i) q^{4} +(-1.02777 + 0.411456i) q^{5} +(2.32125 - 1.49177i) q^{6} +(2.58489 - 0.564210i) q^{7} +(0.654861 + 0.755750i) q^{8} +(-3.62651 + 2.85192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 8 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 4 q^{6} + 15 q^{7} + 16 q^{8} + 28 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{6}{11}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.235759	−	0.971812i	−0.166707	−	0.687175i
							\(3\)	0.902469	+	2.60751i	0.521041	+	1.50545i	0.831863	+	0.554980i	\(0.187274\pi\)
−0.310823	+	0.950468i	\(0.600605\pi\)
\(5\)	−1.02777	+	0.411456i	−0.459631	+	0.184009i	−0.589910	−	0.807469i	\(-0.700837\pi\)
							0.130279	+	0.991477i	\(0.458413\pi\)
\(7\)	2.58489	−	0.564210i	0.976997	−	0.213251i
							\(11\)	−1.27818	+	5.26874i	−0.385386	+	1.58858i	0.367234	+	0.930129i	\(0.380305\pi\)
−0.752620	+	0.658455i	\(0.771210\pi\)
\(13\)	0.378695	−	0.829225i	0.105031	−	0.229986i	−0.849819	−	0.527075i	\(-0.823289\pi\)
							0.954850	+	0.297089i	\(0.0960159\pi\)
\(17\)	−0.303580	−	6.37293i	−0.0736290	−	1.54566i	−0.669639	−	0.742687i	\(-0.733551\pi\)
							0.596010	−	0.802977i	\(-0.296752\pi\)
\(19\)	−0.349058	+	7.32762i	−0.0800793	+	1.68107i	0.499623	+	0.866243i	\(0.333472\pi\)
							−0.579702	+	0.814828i	\(0.696831\pi\)
\(23\)	−2.63278	+	4.00855i	−0.548972	+	0.835841i
							\(29\)	5.10108	−	3.27827i	0.947247	−	0.608759i	0.0268060	−	0.999641i	\(-0.491466\pi\)
0.920441	+	0.390882i	\(0.127830\pi\)
\(31\)	1.03824	−	0.200105i	0.186474	−	0.0359399i	−0.0951592	−	0.995462i	\(-0.530336\pi\)
							0.281633	+	0.959522i	\(0.409124\pi\)
\(37\)	6.90400	−	5.42937i	1.13501	−	0.892583i	0.139785	−	0.990182i	\(-0.455359\pi\)
							0.995226	+	0.0975991i	\(0.0311163\pi\)
\(41\)	−0.268662	+	1.86859i	−0.0419580	+	0.291824i	0.958029	+	0.286672i	\(0.0925490\pi\)
							−0.999987	+	0.00515194i	\(0.998360\pi\)
\(43\)	6.31129	−	7.28361i	0.962462	−	1.11074i	−0.0313326	−	0.999509i	\(-0.509975\pi\)
							0.993795	−	0.111231i	\(-0.0354794\pi\)
\(47\)	−1.63524	−	2.83232i	−0.238524	−	0.413136i	0.721767	−	0.692136i	\(-0.243330\pi\)
							−0.960291	+	0.279000i	\(0.909997\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	322.2.m.a.179.8	yes	160
7.2	even	3	inner	322.2.m.a.317.1	yes	160
23.9	even	11	inner	322.2.m.a.193.1	yes	160
161.9	even	33	inner	322.2.m.a.9.8	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.m.a.9.8	✓	160	161.9	even	33	inner
322.2.m.a.179.8	yes	160	1.1	even	1	trivial
322.2.m.a.193.1	yes	160	23.9	even	11	inner
322.2.m.a.317.1	yes	160	7.2	even	3	inner